scholarly journals Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

2019 ◽  
Vol 871 ◽  
pp. 1028-1043
Author(s):  
M. Abid ◽  
C. Kharif ◽  
H.-C. Hsu ◽  
Y.-Y. Chen
Keyword(s):  
Growth Rate ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Capillary Waves ◽  
Two Dimensional ◽  
Transverse Instability ◽  
Constant Vorticity ◽  
Dimensional Gravity ◽  
Wave Properties

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

Two-dimensional gravity–capillary solitary waves on deep water: generation and transverse instability

2017 ◽  
Vol 834 ◽  
pp. 92-124 ◽  
Author(s):  
Beomchan Park ◽  
Yeunwoo Cho
Keyword(s):  
Solitary Waves ◽  
Deep Water ◽  
Three Dimensional ◽  
Phase Speed ◽  
Narrow Slit ◽  
Two Dimensional ◽  
Transverse Instability ◽  
High Speeds ◽  
Linear Waves ◽  
Dimensional Gravity

Two-dimensional (2-D) gravity–capillary solitary waves are generated using a moving pressure jet from a 2-D narrow slit as a forcing onto the surface of deep water. The forcing moves horizontally over the surface of the deep water at speeds close to the minimum phase speed $c_{min}=23~\text{cm}~\text{s}^{-1}$. Four different states are observed according to the forcing speed. At relatively low speeds below $c_{min}$, small-amplitude depressions are observed and they move steadily just below the moving forcing. As the forcing speed increases towards $c_{min}$, nonlinear 2-D gravity–capillary solitary waves are observed, and they move steadily behind the moving forcing. When the forcing speed is very close to $c_{min}$, periodic shedding of a 2-D local depression is observed behind the moving forcing. Finally, at relatively high speeds above $c_{min}$, a pair of short and long linear waves is observed, respectively ahead of and behind the moving forcing. In addition, we observe the transverse instability of free 2-D gravity–capillary solitary waves and, further, the resultant formation of three-dimensional gravity–capillary solitary waves. These experimental observations are compared with numerical results based on a model equation that admits gravity–capillary solitary wave solutions near $c_{min}$. They agree with each other very well. In particular, based on a linear stability analysis, we give a theoretical proof for the transverse instability of the 2-D gravity–capillary solitary waves on deep water.

scholarly journals Stability and dynamics of two-dimensional fully nonlinear gravity–capillary solitary waves in deep water

2016 ◽  
Vol 809 ◽  
pp. 530-552 ◽  
Author(s):  
Z. Wang
Keyword(s):  
Solitary Waves ◽  
Deep Water ◽  
Wave Equations ◽  
Turning Point ◽  
Phase Speed ◽  
Amplitude Curve ◽  
Two Dimensional ◽  
Fully Nonlinear ◽  
Dimensional Gravity ◽  
The Stability

The stability and dynamics of two-dimensional gravity–capillary solitary waves in deep water within the fully nonlinear water-wave equations are numerically studied. It is well known that there are two families of symmetric gravity–capillary solitary waves – depression waves and elevation waves – bifurcating from infinitesimal periodic waves at the minimum of the phase speed. The stability of both branches was previously examined by Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) by means of a numerical spectral analysis. Their results show that the depression solitary waves with single-valued profiles are stable, while the elevation branch experiences a stability exchange at a turning point on the speed–amplitude curve. In the present paper, we provide numerical evidence that the depression solitary waves with an overhanging structure are also stable. On the other hand, Dias et al. (Eur. J. Mech. B, vol. 15, 1996, pp. 17–36) numerically traced the elevation branch and discovered that its speed–amplitude bifurcation curve features a ‘snake-like’ behaviour with many turning points, whereas Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) only considered the stability exchange near the first turning point. Our results reveal that the stability exchange occurs again near the second turning point. A branch of asymmetric solitary waves is also considered and found to be unstable, even when the wave profile consists of a depression wave and a stable elevation one. The excitation of stable gravity–capillary solitary waves is carried out via direct numerical simulations. In particular, the stable elevation waves, which feature two troughs connected by a small dimple, can be excited by moving two fully localised, well-separated pressures on the free surface with the speed slightly below the phase speed minimum and removing the pressures simultaneously after a period of time.

Transverse instability of surface solitary waves. Part 2. Numerical linear stability analysis

2010 ◽  
Vol 657 ◽  
pp. 126-170 ◽  
Author(s):  
TAKESHI KATAOKA
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Solitary Wave ◽  
Asymptotic Analysis ◽  
Solitary Waves ◽  
Linear Stability Analysis ◽  
Eigenvalues And Eigenfunctions ◽  
Transverse Instability ◽  
Unstable Surface ◽  
Increasing Function

In a previous work, Kataoka & Tsutahara (J. Fluid Mech., vol. 512, 2004a, p. 211) proved the existence of longitudinally stable but transversely unstable surface solitary waves by asymptotic analysis for disturbances of small transverse wavenumber. In the present paper, the same transverse instability is examined numerically for the whole range of solitary-wave amplitudes and transverse wavenumbers of disturbances. Numerical results show that eigenvalues and eigenfunctions of growing disturbance modes agree well with those obtained by the asymptotic analysis if the transverse wavenumber of the disturbance is small. As the transverse wavenumber increases, however, the growth rate of the disturbance, which is an increasing function for small wavenumbers, reaches a maximum and finally falls to zero at some finite wavenumber. Thus, there is a high-wavenumber cutoff to the transverse instability. For higher amplitude, solitary waves become longitudinally unstable, and the dependence of the eigenvalues on the transverse wavenumber exhibits various complicated patterns. We found that such eigenvalues versus transverse wavenumber can be simply grouped into three basic classes.

Comment on ‘Propagation of solitary waves and shock wavelength in the pair plasma (J. Plasma Phys. 78, 525–529, 2012)’

2014 ◽  
Vol 80 (3) ◽  
pp. 513-516
Author(s):  
Frank Verheest
Keyword(s):  
Shock Wave ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Plasma Phys ◽  
Theoretical Underpinning ◽  
Pair Plasma ◽  
Electrons And Positrons ◽  
Small Dissipation ◽  
Pseudopotential Approach ◽  
Wave Properties

In a recent paper ‘Propagation of solitary waves and shock wavelength in the pair plasma (J. Plasma Phys. 78, 525–529, 2012)’, Malekolkalami and Mohammadi investigate nonlinear electrostatic solitary waves in a plasma comprising adiabatic electrons and positrons, and a stationary ion background. The paper contains two parts: First, the solitary wave properties are discussed through a pseudopotential approach, and then the influence of a small dissipation is intuitively sketched without theoretical underpinning. Small dissipation is claimed to lead to a shock wave whose wavelength is determined by linear oscillator analysis. Unfortunately, there are errors and inconsistencies in both the parts, and their combination is incoherent.

Effects of plane progressive irrotational waves on thermal boundary layers

1971 ◽  
Vol 50 (2) ◽  
pp. 321-334 ◽  
Author(s):  
James Witting
Keyword(s):  
Boundary Layers ◽  
Deep Water ◽  
Maximum Amplitude ◽  
Capillary Waves ◽  
Temperature Gradients ◽  
Two Dimensional ◽  
Inviscid Incompressible Fluid ◽  
Mean Temperature ◽  
The Waves ◽  
Thermal Boundary Layers

The average changes in the structure of thermal boundary layers at the surface of bodies of water produced by various types of surface waves are computed. the waves are two-dimensional plane progressive irrotational waves of unchanging shape. they include deep-water linear waves, deep-water capillary waves of arbitrary amplitude, stokes waves, and the deep-water gravity wave of maximum amplitude.The results indicate that capillary waves can decrease mean temperature gradients by factors of as much as 9·0, if the average heat flux at the air-water interface is independent of the presence of the waves. Irrotational gravity waves can decrease the mean temperature gradients by factors no more than 1·381.Of possible pedagogical interest is the simplicity of the heat conduction equation for two-dimensional steady irrotational flows in an inviscid incompressible fluid if the velocity potential and the stream function are taken to be the independent variables.

Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions

2009 ◽  
Vol 465 (2109) ◽  
pp. 2725-2749 ◽  
Author(s):  
M. J. Ablowitz ◽  
T. S. Haut
Keyword(s):  
Surface Tension ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Asymptotic Series ◽  
High Order ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Small Surface ◽  
Petviashvili Equation ◽  
Dimensional Series

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

scholarly journals Dynamics of steep two-dimensional gravity–capillary solitary waves

2010 ◽  
Vol 664 ◽  
pp. 466-477 ◽  
Author(s):  
PAUL A. MILEWSKI ◽  
J.-M. VANDEN-BROECK ◽  
ZHAN WANG
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Infinite Depth ◽  
Linear Dispersion Relation ◽  
Dimensional Gravity ◽  
Nonlinear Free Surface ◽  
The Waves

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

Sign in / Sign up

Export Citation Format

Share Document